Here they are:
1. It Helps You Learn Math
The best way to learn something is to teach it! An oversimplification, perhaps, but teaching is a definitely an awesome way to learn something. Writing a blog can also help you be more careful and precise in your learning because any mistake will be displayed to the world. Occasionally I've even looked something up on this blog because I forgot it!
2. It's Fun to Publish Something
Let's face it, research papers take a long time to write and it takes a long time to publish. In fact, a year of waiting after submission is pretty normal in math. It's fun to see something you write be published immediately, even if it's not something new. And, it's really nice to receive comments.
3. It Improves Your Writing
The practice of being concise and clear are things I've enjoyed thoroughly through blogging. Writing a blog is a great way to practice for other types of writing, which is a pretty important skill to have.
4. It Can Make You Connections
I've actually had occasional math discussions with people as a direct result of blogging!
5. It's a Good Distraction from Research
Most of what I write about here is not about my research, and for good reason. In fact, I make it a point to blog mainly about random things that I find interesting rather than my directed research program. Even so, often thoughts that intially seemed to be disjoint from my research have given me new ideas.
The point is, it's good to have a mathematical distraction from a sometimes myopic research goal to broaden your horizons, give your brain a break from intense direction, and spark new ideas.
And, speaking of blogging, I've started a new experimental blog called Random Noise. It's a blog about photography. Check it out! Please note, however, that Aleph Zero Categorical won't go the way of the dodo just because I have a new blog.